0

I was asked to prove the following : Let $G,H,$ and $K$ be finite abelian groups. If $G \times H \cong G\times K$ then $H\cong K$. It seemed very intuitive to me but I am having a hard time writing it formally. I found this : Finite Abelian groups: $G \times H \cong G\times K$ then $H\cong K$ which is the same question but the explanation there is not really formal and I feel as if they only reduced the problem to an equal one: 1. first me decompose to cyclic groups 2.we say that G is of course decomposed in the same way in the two products 3. since the decomposition are isomorphic and G's decomposition is the same we conclude that K's decomposition is isomorphic to H's decomposition. But isn't it the same as saying the same argument without decomposing at all? What am I missing here? What does the decomposition really gives us apart from writing the same thing in a different way?

Community
  • 1
user2129387
  • 31
  • What would be the argument without decomposing at all, that you are thinking of? – Sarvesh Ravichandran Iyer Sep 23 '16 at 09:41
  • since the two product are isomorphic and the left side is isomorphic as well the right side has to be isomorphic as well. I know it is wrong but it seems to me as that is exactly the same argument as is used after decomposing, but for some reason there it is legit. – user2129387 Sep 23 '16 at 09:46
  • The best way to convince yourself that your argument is wrong, is to find a counterexample with non-abelian groups. – Sarvesh Ravichandran Iyer Sep 23 '16 at 09:47
  • I know it is wrong ! but how does it defer from the argument they use after decomposing? – user2129387 Sep 23 '16 at 09:48
  • 5
    Hang on. I think this fact is true for all finite groups (abelian or non-abelian), and there is an argument not requiring group theory at all. Check some of the "linked" questions other than the one you posted. You will find interesting arguments there, like : http://math.stackexchange.com/questions/349855/does-g-times-k-cong-h-times-k-imply-g-cong-h?noredirect=1&lq=1 – Sarvesh Ravichandran Iyer Sep 23 '16 at 10:09
  • I saw it too, but it seems pretty advanced and I would like to know how to prove this with the tool I am familiar with right now. – user2129387 Sep 23 '16 at 10:35
  • I unfortunately cannot help then, because I am only a group theory beginner. In truth I did not even know what the fundamental theorem for finite abelian groups was before reading your post. – Sarvesh Ravichandran Iyer Sep 23 '16 at 10:36

0 Answers0